A Polynomial-Structured Encoding Method for High-Density QC-LDPC Codes

High-density (HD) quasi-cyclic low-density parity-check (QC-LDPC) codes are widely adopted in high-speed, high-reliability optical communication systems. However, the high density of the quasi-cyclic parity-check matrix prevents the direct derivation of a corresponding quasi-cyclic generator matrix, leading to computationally prohibitive encoding complexity. To address this limitation, based on the established polynomial-ring representation of QC-LDPC codes, this paper develops a structure-preserving polynomial-domain transformation for the high-density 50G-PON QC-LDPC parity-check matrix. The proposed method transforms the dense quasi-cyclic parity-check matrix into a compact systematic encoding form over $R={\mathbb{F}}_2\left[x\right]/(x^{256}-1)$. As a result, parity generation is reduced to the inversion of a small $3\times 3$ polynomial submatrix and a sequence of cyclic-shift-and-XOR operations. Based on this construction, an optimized HD-QC-LDPC encoding algorithm and its corresponding FPGA architecture are developed. The resulting hardware encoder achieves a throughput of 58.9 Gbps at a 200 MHz clock frequency on a Xilinx Kintex-7 FPGA, satisfying the throughput and latency requirements of 50G-PON systems.